5Th Lecture : Modular Decomposition MPRI 2013–2014 Schedule A
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Odd Harmonious Labeling on Pleated of the Dutch Windmill Graphs
CAUCHY – JURNAL MATEMATIKA MURNI DAN APLIKASI Volume 4 (4) (2017), Pages 161-166 p-ISSN: 2086-0382; e-ISSN: 2477-3344 Odd Harmonious Labeling on Pleated of the Dutch Windmill Graphs Fery Firmansah1, Muhammad Ridlo Yuwono2 1, 2Mathematics Edu. Depart. University of Widya Dharma Klaten, Indonesia Email: [email protected], [email protected] ABSTRACT A graph 퐺(푉(퐺), 퐸(퐺)) is called graph 퐺(푝, 푞) if it has 푝 = |푉(퐺)| vertices and 푞 = |퐸(퐺)| edges. The graph 퐺(푝, 푞) is said to be odd harmonious if there exist an injection 푓: 푉(퐺) → {0,1,2, … ,2푞 − 1} such that the induced function 푓∗: 퐸(퐺) → {1,3,5, … ,2푞 − 1} defined by 푓∗(푢푣) = 푓(푢) + 푓(푣). The function 푓∗ is a bijection and 푓 is said to be odd harmonious labeling of 퐺(푝, 푞). In this paper we prove that pleated of the (푘) Dutch windmill graphs 퐶4 (푟) with 푘 ≥ 1 and 푟 ≥ 1 are odd harmonious graph. Moreover, we also give odd (푘) (푘) harmonious labeling construction for the union pleated of the Dutch windmill graph 퐶4 (푟) ∪ 퐶4 (푟) with 푘 ≥ 1 and 푟 ≥ 1. Keywords: odd harmonious labeling, pleated graph, the Dutch windmill graph INTRODUCTION In this paper we consider simple, finite, connected and undirected graph. A graph 퐺(푝, 푞) with 푝 = |푉(퐺)| vertices and 푞 = |퐸(퐺)| edges. A graph labeling which has often been motivated by practical problems is one of fascinating areas of research. Labeled graphs serves as useful mathematical models for many applications in coding theory, communication networks, and mobile telecommunication system. -
Strong Edge Graceful Labeling of Windmill Graphs ∑
International Journal of Mathematics Research. ISSN 0976-5840 Volume 5, Number 1 (2013), pp. 19-26 © International Research Publication House http://www.irphouse.com Strong Edge Graceful Labeling of Windmill Graphs Dr. M. Subbiah VKS College Of Engineering& Technology Desiyamangalam, Karur - 639120 [email protected]. Abstract A (p, q) graph G is said to have strong edge graceful labeling if there 3q exists an injection f from the edge set to 1,2, ... so that the 2 induced mapping f+ defined on the vertex set given by f x fxy xy EG mod 2p are distinct. A graph G is said to be strong edge graceful if it admits a strong edge graceful labeling. In this paper we investigate strong edge graceful labeling of Windmill graph. (n) Definition: The windmill graphs Km (n >3) to be the family of graphs consisting of n copies of Km with a vertex in common. (n) Theorem: 1. The windmill graph K4 is strong edge graceful for all n 3 when n is even. (n) Proof: Let {v1, v2, v3, ..., v3n, } be the vertices of K4 and {e1, e2, e3, ...,e3n- (n) 1, e3n, , f1, ,f2, ,f3, . .f3n-1, f3n. } be the edges of K4 which are denoted as in the following Fig. 1. 20 Dr. M. Subbiah . v e 3 3 n n -1 . -1 . v . 3 n f . 3 n -2 f 3 e n . 3 -1 n e 3 2 n n -2 f v v 3 3n 0 2 v 3 . n-2 f v 1 . 2 f 2 f 3 f . -
Split Decomposition and Graph-Labelled Trees
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector Discrete Applied Mathematics 160 (2012) 708–733 Contents lists available at SciVerse ScienceDirect Discrete Applied Mathematics journal homepage: www.elsevier.com/locate/dam Split decomposition and graph-labelled trees: Characterizations and fully dynamic algorithms for totally decomposable graphsI Emeric Gioan, Christophe Paul ∗ CNRS - LIRMM, Université de Montpellier 2, France article info a b s t r a c t Article history: In this paper, we revisit the split decomposition of graphs and give new combinatorial and Received 20 May 2010 algorithmic results for the class of totally decomposable graphs, also known as the distance Received in revised form 3 February 2011 hereditary graphs, and for two non-trivial subclasses, namely the cographs and the 3-leaf Accepted 23 May 2011 power graphs. Precisely, we give structural and incremental characterizations, leading to Available online 23 July 2011 optimal fully dynamic recognition algorithms for vertex and edge modifications, for each of these classes. These results rely on the new combinatorial framework of graph-labelled Keywords: trees used to represent the split decomposition of general graphs (and also the modular Split decomposition Distance hereditary graphs decomposition). The point of the paper is to use bijections between the aforementioned Fully dynamic algorithms graph classes and graph-labelled trees whose nodes are labelled by cliques and stars. We mention that this bijective viewpoint yields directly an intersection model for the class of distance hereditary graphs. ' 2012 Published by Elsevier B.V. 1. Introduction The 1-join composition and its complementary operation, the split decomposition, range among the classical operations in graph theory. -
Math.RA] 25 Sep 2013 Previous Paper [3], Also Relying in Conceptually Separated Tools from Them, Such As Graphs and Digraphs
Certain particular families of graphicable algebras Juan Núñez, María Luisa Rodríguez-Arévalo and María Trinidad Villar Dpto. Geometría y Topología. Facultad de Matemáticas. Universidad de Sevilla. Apdo. 1160. 41080-Sevilla, Spain. [email protected] [email protected] [email protected] Abstract In this paper, we introduce some particular families of graphicable algebras obtained by following a relatively new line of research, ini- tiated previously by some of the authors. It consists of the use of certain objects of Discrete Mathematics, mainly graphs and digraphs, to facilitate the study of graphicable algebras, which are a subset of evolution algebras. 2010 Mathematics Subject Classification: 17D99; 05C20; 05C50. Keywords: Graphicable algebras; evolution algebras; graphs. Introduction The main goal of this paper is to advance in the research of a novel mathematical topic emerged not long ago, the evolution algebras in general, and the graphicable algebras (a subset of them) in particular, in order to obtain new results starting from those by Tian (see [4, 5]) and others already obtained by some of us in a arXiv:1309.6469v1 [math.RA] 25 Sep 2013 previous paper [3], also relying in conceptually separated tools from them, such as graphs and digraphs. Concretely, our goal is to find some particular types of graphicable algebras associated with well-known types of graphs. The motivation to deal with evolution algebras in general and graphicable al- gebras in particular is due to the fact that at present, the study of these algebras is very booming, due to the numerous connections between them and many other branches of Mathematics, such as Graph Theory, Group Theory, Markov pro- cesses, dynamic systems and the Theory of Knots, among others. -
Applications of Lexicographic Breadth-First Search to Modular Decomposition, Split Decomposition, and Circle Graphs by Marc Tedd
Applications of Lexicographic Breadth-First Search to Modular Decomposition, Split Decomposition, and Circle Graphs by Marc Tedder A thesis submitted in conformity with the requirements for the degree of Doctor of Philosophy Graduate Department of Computer Science University of Toronto Copyright c 2011 by Marc Tedder Abstract Applications of Lexicographic Breadth-First Search to Modular Decomposition, Split Decomposition, and Circle Graphs Marc Tedder Doctor of Philosophy Graduate Department of Computer Science University of Toronto 2011 This thesis presents the first sub-quadratic circle graph recognition algorithm, and develops im- proved algorithms for two important hierarchical decomposition schemes: modular decomposition and split decomposition. The modular decomposition algorithm results from unifying two dif- ferent approaches previously employed to solve the problem: divide-and-conquer and factorizing permutations. It runs in linear-time, and is straightforward in its understanding, correctness, and implementation. It merely requires a collection of trees and simple traversals of these trees. The split-decomposition algorithm is similar in being straightforward in its understanding and correctness. An efficient implementation of the algorithm is described that uses the union-find data-structure. A novel charging argument is used to prove the running-time. The algorithm is the first to use the recent reformulation of split decomposition in terms of graph-labelled trees. This facilitates its extension to circle graph recognition. In particular, it allows us to efficiently apply a new lexicographic breadth-first search characterization of circle graphs developed in the thesis. Lexicographic breadth-first search is additionally responsible for the efficiency of the split decom- position algorithm, and contributes to the simplicity of the modular decomposition algorithm. -
Algorithms for Generating Star and Path of Graphs Using BFS Dr
Web Site: www.ijettcs.org Email: [email protected], [email protected] Volume 1, Issue 3, September – October 2012 ISSN 2278-6856 Algorithms for Generating Star and Path of Graphs using BFS Dr. H. B. Walikar2, Ravikumar H. Roogi1, Shreedevi V. Shindhe3, Ishwar. B4 2,4Prof. Dept of Computer Science, Karnatak University, Dharwad, 1,3Research Scholars, Dept of Computer Science, Karnatak University, Dharwad Abstract: In this paper we deal with BFS algorithm by 1.8 Diamond Graph: modifying it with some conditions and proper labeling of The diamond graph is the simple graph on nodes and vertices which results edges illustrated Fig.7. [2] and on applying it to some small basic 1.9 Paw Graph: class of graphs. The BFS algorithm has to modify The paw graph is the -pan graph, which is also accordingly. Some graphs will result in and by direct application of BFS where some need modifications isomorphic to the -tadpole graph. Fig.8 [2] in the algorithm. The BFS algorithm starts with a root vertex 1.10 Gem Graph: called start vertex. The resulted output tree structure will be The gem graph is the fan graph illustrated Fig.9 [2] in the form of Structure or in Structure. 1.11 Dart Graph: Keywords: BFS, Graph, Star, Path. The dart graph is the -vertex graph illustrated Fig.10.[2] 1.12 Tetrahedral Graph: 1. INTRODUCTION The tetrahedral graph is the Platonic graph that is the 1.1 Graph: unique polyhedral graph on four nodes which is also the A graph is a finite collection of objects called vertices complete graph and therefore also the wheel graph . -
Interlace Polynomials of Friendship Graphs
Electronic Journal of Graph Theory and Applications 6 (2) (2018), 269–281 Interlace polynomials of friendship graphs Christina Eubanks-Turnera, Aihua Lib aDepartment of Mathematics, Loyola Marymount University, 90045 bDepartment of Mathematical Sciences, Montclair State University, 07043 [email protected], [email protected] Abstract In this paper, we study the interlace polynomials of friendship graphs, that is, graphs that satisfy the Friendship Theorem given by Erdos,¨ Renyi´ and Sos. Explicit formulas, special values, and behaviour of coefficients of these polynomials are provided. We also give the interlace polynomials of other similar graphs, such as, the butterfly graph. Keywords: graph polynomial, interlace polynomial, friendship graph, butterfly graph Mathematics Subject Classification : 05C31, 05C50 DOI: 10.5614/ejgta.2018.6.2.7 1. Introduction Sequencing by hybridization is a method of reconstructing a long DNA string from its nu- cleotide sequence. Since gaining a unique reconstruction from the substrings is not always possi- ble, a major question that arises in this study is “For a random string, how many reconstructions are possible?” In [2], Arratia, Bollobas,´ Coppersmith, and Sorkin answer an important question related to DNA sequencing by converting this to a question about Euler circuits in a 2-in, 2-out graph that have been “toggled” (interlaced). The previously mentioned authors introduced the in- terlace polynomial of a graph, a polynomial that represents the information gained from doing the toggling process on the graph, see [2]. Interlace polynomials are similar to other graph polynomi- als, such as, Tutte and Martin polynomials, see [5]. Some researchers have studied different types of graph polynomials, such as genus polynomials, [7]. -
The Use of a Pruned Modular Decomposition for Maximum Matching Algorithms on Some Graph Classes
The Use of a Pruned Modular Decomposition for Maximum Matching Algorithms on Some Graph Classes Guillaume Ducoffe ICI – National Institute for Research and Development in Informatics, Bucharest, Romania The Research Institute of the University of Bucharest ICUB, Bucharest, Romania guillaume.ducoff[email protected] Alexandru Popa University of Bucharest, Bucharest, Romania ICI – National Institute for Research and Development in Informatics, Bucharest, Romania [email protected] Abstract We address the following general question: given a graph class C on which we can solve Maximum Matching in (quasi) linear time, does the same hold true for the class of graphs that can be modularly decomposed into C? As a way to answer this question for distance-hereditary graphs and some other superclasses of cographs, we study the combined effect of modular decomposition with a pruning process over the quotient subgraphs. We remove sequentially from all such subgraphs their so-called one-vertex extensions (i.e., pendant, anti-pendant, twin, universal and isolated vertices). Doing so, we obtain a “pruned modular decomposition”, that can be computed in quasi linear time. Our main result is that if all the pruned quotient subgraphs have bounded order then a maximum matching can be computed in linear time. The latter result strictly extends a recent framework in (Coudert et al., SODA’18). Our work is the first to explain why the existence of some nice ordering over the modules of a graph, instead of just over its vertices, can help to speed up the computation of maximum matchings on some graph classes. 2012 ACM Subject Classification Mathematics of computing → Graph theory, Theory of com- putation → Design and analysis of algorithms Keywords and phrases maximum matching, FPT in P, modular decomposition, pruned graphs, one-vertex extensions, P4-structure Digital Object Identifier 10.4230/LIPIcs.ISAAC.2018.6 Related Version A full version of the paper is available at [14], https://arxiv.org/abs/1804. -
On the Chromatic Number of (P {5},Windmill)-Free Graphs
Opuscula Math. 37, no. 4 (2017), 609–615 http://dx.doi.org/10.7494/OpMath.2017.37.4.609 Opuscula Mathematica ON THE CHROMATIC NUMBER OF (P5, windmill)-FREE GRAPHS Ingo Schiermeyer Communicated by Hao Li Abstract. In this paper we study the chromatic number of (P5, windmill)-free graphs. For p integers r, p 2 the windmill graph W = K pKr is the graph obtained by joining a single ≥ r+1 1 ∨ vertex (the center) to the vertices of p disjoint copies of a complete graph Kr. Our main result is that every (P5, windmill)-free graph G admits a polynomial χ-binding function. Moreover, we will present polynomial χ-binding functions for several other subclasses of P5-free graphs. Keywords: vertex colouring, perfect graphs, χ-binding function, forbidden induced subgraph. Mathematics Subject Classification: 05C15, 05C17. 1. INTRODUCTION We consider finite, simple, and undirected graphs, and use standard terminology and notation. Let G be a graph. An induced subgraph of G is a graph H such that V (H) V (G), ⊆ and uv E(H) if and only if uv E(G) for all u, v V (H). Given graphs G and F ∈ ∈ ∈ we say that G contains F if F is isomorphic to an induced subgraph of G. We say that a graph G is F -free, if it does not contain F. For two graphs G, H we denote by G + H the disjoint union and by G H the join of G and H, respectively. ∨ A graph G is called k-colourable, if its vertices can be coloured with k colours so that adjacent vertices obtain distinct colours. -
Regular and Biregular Planar Cages
REGULAR AND BIREGULAR PLANAR CAGES GABRIELA ARAUJO-PARDO 1, FIDEL BARRERA-CRUZ 2, AND NATALIA GARC´IA-COL´IN 3 Abstract. We study the Cage Problem for regular and biregular planar graphs. A (k; g)-graph is a k-regular graph with girth g.A(k; g)-cage is a (k; g)-graph of minimum order. It is not difficult to conclude that the regu- lar planar cages are the Platonic Solids.A(fr; mg; g)-graph is a graph of girth g whose vertices have degrees r and m: A(fr; mg; g)-cage is a (fr; mg; g)-graph of minimum order. In this case we determine the triplets of values (fr; mg; g) for which there exist planar (fr; mg; g){graphs, for all those values we construct examples. Furthermore, for many triplets (fr; mg; g) we build the (fr; mg; g)- cages. Keywords: Cages, Planar Graphs. MSC2010: 05C35, 05C10. 1. Introduction We only consider finite simple graphs. The girth of a graph is the length of a smallest cycle. A (k; g)-graph is a k-regular graph with girth g.A(k; g)-cage is a (k; g)-graph of minimum order, n(k; g): These graphs were introduced by Tutte in 1947 (see [14]). The Cage Problem consists of finding the (k; g)-cages for any pair integers k ≥ 2 and g ≥ 3: However, this challenge has proven to be very difficult even though the existence of (k; g)-graphs was proved by Erd¨osand Sachs in 1963 (see [10]). There is a known natural lower bound for the order of a cage, called Moore's lower bound and denoted by n0(r; g): It is obtained by counting the vertices of a rooted tree, T(g−1)=2 with radius (g − 1)=2, if g is odd; or the vertices of a \double-tree" rooted at an edge (that is, two different rooted trees T(g−3)=2 with the root vertices incident to an edge) if g is even (see [9, 11]). -
Fully Dynamic Algorithm for Recognition and Modular Decomposition of Permutation Graphs
Algorithmica DOI 10.1007/s00453-008-9273-0 Fully Dynamic Algorithm for Recognition and Modular Decomposition of Permutation Graphs Christophe Crespelle · Christophe Paul Received: 5 June 2006 / Accepted: 22 December 2008 © Springer Science+Business Media, LLC 2009 Abstract This paper considers the problem of maintaining a compact representa- tion (O(n) space) of permutation graphs under vertex and edge modifications (inser- tion or deletion). That representation allows us to answer adjacency queries in O(1) time. The approach is based on a fully dynamic modular decomposition algorithm for permutation graphs that works in O(n) time per edge and vertex modification. We thereby obtain a fully dynamic algorithm for the recognition of permutation graphs. Keywords Dynamic algorithms · Permutation graphs · Modular decomposition 1 Introduction Finding efficient graph representations is a central question of algorithmic graph the- ory. How to store a graph to make its manipulation easier? Compact graph represen- tations often rest on the combinatorial structures of the considered graphs (see for example [22]). Thereby interesting graph encodings can be found when restricted to special graph classes. This paper deals with dynamic graphs and considers the dy- namic recognition and representation problem (see e.g. [21]), which, for a family F of graphs, aims to maintain a characteristic representation of dynamically changing graphs as long as the modified graph belongs to F. The input of the problem is a graph G ∈ F with its representation and a series of modifications. Any modification This paper is a full version of the extended abstract appeared in [5]. C. Crespelle () Université de Montpellier 2, LIRMM, Montpellier, France e-mail: [email protected] C. -
Optimal Information Ratio of Secret Sharing Schemes on Dutch Windmill Graphs
Advances in Mathematics of Communications doi:10.3934/amc.2019005 Volume 13, No. 1, 2019, 89–99 OPTIMAL INFORMATION RATIO OF SECRET SHARING SCHEMES ON DUTCH WINDMILL GRAPHS Bagher Bagherpour∗, Shahrooz Janbaz and Ali Zaghian Applied Mathematics and Cryptography Department Malek Ashtar university of technology, Isfahan, Iran (Communicated by Jens Zumbraegel) Abstract. One of the basic problems in secret sharing is to determine the exact values of the information ratio of the access structures. This task is important from the practical point of view, since the security of any system degrades as the amount of secret information increases. A Dutch windmill graph consists of the edge-disjoint cycles such that all of them meet in one vertex. In this paper, we determine the exact information ratio of secret sharing schemes on the Dutch windmill graphs. Furthermore, we determine the exact ratio of some related graph families. 1. Introduction Let P = {p1,p2,...,pn} be the set of participants among which the dealer wants to share some secret s in such a way that only the qualified subsets of P can reconstruct the secret s. A secret sharing scheme is called perfect if the non-qualified subsets of P can not obtain any information about the secret s. 2P denotes the set of all subsets of the set P , and Γ is a collection of subsets of P . We say that Γ is monotone over P if A ∈ Γ and A ⊆ A′, then A′ ∈ Γ. In the secret sharing schemes, the access structure Γ over P is a collection of all qualified subsets of P that is monotone and ∅ ∈/ Γ.